Is there and existing Blender renderer REST API service?
Perhaps you should take a look at RenderStreet.
I used it for my work as well, what i like about it is that they provide a plugin in Blender, from which you can upload a blend file. Then they do the rendering, they do an estimate of the cost (which can change over time, since your render frames can get more complex); you can ask someone else to pay ahead your render time (ea your manager, or your custommer). And they have web interface that looks quite well on a mobile phone. you can pause / start (multiple) jobs, and they have also FTP access to download your end result. 

You might get cheaper elsewhere, but the fact that they offer such billing, and preview made it very easy to combine with real work.
Their support is also quite good, and their speed is blazing.

For sure there other render farms, and even amazone scripts.
But in a work situation this might be as close to saas in a "company-sense-work" way as one might get. 

And if you a little bit more time left, you might find an old Coin minner who sells his hardware (that's my current situation), i ussually use that now unless i have to finish something in one day, then i go to a render farm solution.

(oh you might edit the script of renderstreet to make it even more Rest Alike, or ask their support to create it for you).